Note that we do not specify the local geometry of \(\mathcal {M}\) , but only the asymptotic one at the ends. This allows
for very different behaviors of the geodesics. For the case of the one-sheeted hyperboloid, for example, the geodesic flow
around the unique periodic geodesic is hyperbolic in the sense of dynamical systems, whereas if we place a section of \(S^2\) 
in the middle of \(\mathcal {M}\)  then we encounter a set of positive measure in the cotangent bundle leading to stable periodic geodesics.
These two scenarios are depicted in Figure 2.
It is natural to ask to what extent this local geometry affects the dispersion of the flow.
The following result summarizes what is proved in [1]}, [2]}
for the case of \(\Omega =S^1\)  (see those references for general compact \(\Omega \) ).
